PHY 711 Fall 2014 -- Lecture 34 111/14/2014

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 34:

Chapter 10 in F & W: Surface waves

-- Non-linear contributions and soliton solutions

PHY 711 Fall 2014 -- Lecture 34 211/14/2014

PHY 711 Fall 2014 -- Lecture 3411/14/2014 3

p0

hz

x

z

y

212

2

Within fluid: 0

constant

0

( , , , )

( , , , ( , , , )

At surface: with

)

z h

v g z ht

x y z

x y z t

xt

z h

y z t

v v

, ,

, ,

where , , ,x y x y x y

x y t

dv v v v x y h t

dt t x y

General problem including non-linearities

Surface waves in an incompressible fluid

PHY 711 Fall 2014 -- Lecture 3411/14/2014 4

p0

hz

x

z

y

dt

d

zthzxv

hz

txtzx

y

z

,, :surfaceAt

0 :fluidWithin

, ,,

dependence- trivialassume tions;simplificaFurther

PHY 711 Fall 2014 -- Lecture 34 511/14/2014

p0

hz

x

z

yz=0

2 00 0

0

0

Dominant non-linear effects soliton solutions

3( , ) sech

2

where 11 / 2

constantx ct

x th h

ghc gh

h h

Non-linear effects in surface waves:

PHY 711 Fall 2014 -- Lecture 34 611/14/2014

Detailed analysis of non-linear surface waves[Note that these derivations follow Alexander L. Fetter and John Dirk Walecka, Theoretical Mechanics of Particles and Continua (McGraw Hill, 1980), Chapt. 10.]

We assume that we have an incompressible fluid: constant

Velocity potential: ( , , ); ( , , ) ( , , )x z t x z t x z t

v

The surface of the fluid is described by z=h+z(x,t). It is assumed that the fluid is contained in a structure (lake, river, swimming pool, etc.) with a structureless bottom defined by the z = 0 plane and filled to an equilibrium height of z = h.

PHY 711 Fall 2014 -- Lecture 34 711/14/2014

Defining equations for F(x,z,t) and z(x,t)

where 0 ( , )z h x t

2 2

2 2

( , , ) ( , , )0 0

Continuity equation:

x z t x z t

x z

v

Bernoulli equation (assuming irrotational flow) and gravitation potential energy

2 2( , , ) 1 ( , , ) ( , , )

( ) 0.2

x z t x z t x z tg z h

t x z

PHY 711 Fall 2014 -- Lecture 34 811/14/2014

Boundary conditions on functions –

Zero velocity at bottom of tank:

( ,0, )0.

x t

z

Consistent vertical velocity at water surface

( , , ) . z z h

dv x z t

dt t

vt

( , , ) ( , , ) ( , ) ( , )0z h

x z t x z t x t x t

z x x t

t

PHY 711 Fall 2014 -- Lecture 34 911/14/2014

Analysis assuming water height z is small relative tovariations in the direction of wave motion (x)Taylor’s expansion about z = 0:

2 2 3 3 4 4

2 3 4( , , ) ( ,0, ) ( ,0, ) ( ,0, ) ( ,0, ) ( ,0, )

2 3! 4!

z z zx z t x t z x t x t x t x t

z z z z

Note that the zero vertical velocity at the bottom ensures that all odd derivatives vanish from the

Taylor expansion . In addition, the Laplace equation allows us to convert all even derivatives with respect to zto derivatives with respect to x.

( ,0, )n

nx t

z

2 2 4 4

2 4Modified Taylor s expansion: ( , , ) ( ,0, ) ( ,0, ) ( ,0, )

2 4!

z zx z t x t x t x t

x x

PHY 711 Fall 2014 -- Lecture 34 1011/14/2014

Check linearized equations and their solutions: Bernoulli equations --

Bernoulli equation evaluated at ( , )

( , , )( , ) 0

Consistent vertical velocity at ( , )

( , , ) ( , )0z h

z h x t

x h tg x t

tz h x t

x z t x t

z t

t

Using Taylor's expansion results to lowest order

2

2

( , , ) ( ,0, ) ( , , ) ( ,0, )x z t x t x h t x th

z x t t

2 2

2 2

( ,0Decoupl

, )ed equations:

( ,0, ).

x t x tgh

t x

PHY 711 Fall 2014 -- Lecture 34 1111/14/2014

Analysis of non-linear equations -- keeping the lowest order nonlinear terms and include up to 4th order derivatives in the linear terms. Let ( , ) ( ,0, )x t x t

222 3 2

2 2

22 3

2

Approximate form of Bernoulli equation evaluated at surface:

( ) 1( ) 0

2 2

10.

2 2

z h

hh g

t t x x x

hg

t t x x

3 4

4

Approximate form of surface velocity expressio

( ( , )) 0.3!

n :

hh x t

x x x t

The expressions keep the lowest order nonlinear terms and include up to 4th order derivatives in the linear terms.

PHY 711 Fall 2014 -- Lecture 34 1211/14/2014

22 3

2

3 4

4

10.

2 2

Cou

( ( ,

pled

)) 0.

equation

!

s:

3

hg

t t x x

hh x t

x x x t

Traveling wave solutions with new nota

t

( , ) ( ) and ( , ) (

io

)

n:

u x ct x t u x t u

22 3

3

3 4

4

( ) ( ) 1 ( )( ) 0.

2 2

( ) ( ) ( )( ( )) 0.

6

d u ch d u d uc g udu du du

d d u h d u d uh u c

du du du du

Note that the wave “speed” c will be consistently determined

PHY 711 Fall 2014 -- Lecture 34 1311/14/2014

Integrating and re-arranging coupled equations

22 3

3

2 2 22 2

3

3 4

4

3

( ) ( ) 1 ( )( ) 0.

2 2

1' ''' ( ') ''

2 2 2 2

( ) ( ) ( )( ( )) 0.

6

( ) ' ''' 06

d u ch d u d uc g udu du du

g h g h g g

c c c c c

d d u h d u d uh u c

du du du du

hh c

PHY 711 Fall 2014 -- Lecture 34 1411/14/2014

Integrating and re-arranging coupled equations – continued --Expressing modified surface velocity equation in terms of h(u):

2Note: c gh

2 2 32

3

32

2 2 2 2

22

2

( ) '' '' 02 2 6

1 '' 1 03 2

31 ( ) ''( ) ( ) 0.

3 2

g h g g h gh c

c c c c

gh gh g gh

c c c c

hg hu u u

c h

PHY 711 Fall 2014 -- Lecture 34 1511/14/2014

Solution of the famous Korteweg-de Vries equation

Modified surface amplitude equation in terms of h

2

2

2

31 ( ) ''( ) ( ) 0.

3 2

hg hu u u

c h

Soliton solution

2 00

00

0

is a constan

3( , ) ( ) sech

2

1 where 1 / 2

t

x ctx t x ct

h h

ghc gh

h h

PHY 711 Fall 2014 -- Lecture 34 1611/14/2014

Relationship to “standard” form of Korteweg-de Vries equation

00

3 32 , , and .

2 2 2

New variables:

x ctx t

h h h h

3

3

Standard Korteweg-de Vries eq

6 0

i n

.

uat o

t x x

2( , ) sech ( ) .

Soliton soluti

2

on:

2x t x t

PHY 711 Fall 2014 -- Lecture 34 1711/14/2014

More details

2

2

2

02

20

Modified surface amplitude equation in terms of :

31 ( ) ( ) ( ) 0.

3 2

Some identities: 1 ; ; .

Derivative of surface amplitude equation:

3''

3' ' '

hg hu u u

c h

gh d dc

h c t du x du

h

h h

2 30

3

3

3

0.

Expression in terms of and :

30.

3Expression in terms of and :

6 0.

x t

h

ch t x h xx t

t x x

PHY 711 Fall 2014 -- Lecture 34 1811/14/2014

Soliton solution

2 00

00

0

is a constan

3( , ) ( ) sech

2

1 where 1 / 2

t

x ctx t x ct

h h

ghc gh

h h

Summary

PHY 711 Fall 2014 -- Lecture 34 1911/14/2014

Some links: Website – http://www.ma.hw.ac.uk/solitons/

Photo of canal soliton http://www.ma.hw.ac.uk/solitons/